effective field theory of weakly coupled inflationary models
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Effective field theory of weakly coupled inflationary models
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JCAP04(2013)004
ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ
Effective field theory of weakly
coupled inflationary models
Rhiannon Gwyn,a Gonzalo A. Palma,b Mairi Sakellariadouc andSpyros Sypsasc
aMax-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut,Am Muhlenberg 1, D-14476 Potsdam, Germany
bPhysics Department, FCFM, Universidad de Chile,Blanco Encalada 2008, Santiago, Chile
cDepartment of Physics, King’s College London,Strand, London WC2R 2LS, U.K.
E-mail: [email protected], [email protected],[email protected], [email protected]
Received October 16, 2012Accepted March 12, 2013Published April 2, 2013
Abstract. The application of Effective Field Theory (EFT) methods to inflation has takena central role in our current understanding of the very early universe. The EFT perspectivehas been particularly useful in analyzing the self-interactions determining the evolution ofco-moving curvature perturbations (Goldstone boson modes) and their influence on low-energy observables. However, the standard EFT formalism, to lowest order in spacetimedifferential operators, does not provide the most general parametrization of a theory thatremains weakly coupled throughout the entire low-energy regime. Here we study the EFTformulation by including spacetime differential operators implying a scale dependence of theGoldstone boson self-interactions and its dispersion relation. These operators are shown toarise naturally from the low-energy interaction of the Goldstone boson with heavy fieldsthat have been integrated out. We find that the EFT then stays weakly coupled all theway up to the cutoff scale at which ultraviolet degrees of freedom become operative. Thisopens up a regime of new physics where the dispersion relation is dominated by a quadraticdependence on the momentum ω ∼ p2. In addition, provided that modes crossed the Hubblescale within this energy range, the predictions of inflationary observables — including non-Gaussian signatures — are significantly affected by the new scales characterizing it.
Keywords: inflation, physics of the early universe, cosmological perturbation theory
ArXiv ePrint: 1210.3020
c© 2013 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2013/04/004
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Contents
1 Introduction & summary 1
2 Review of the standard EFT formalism 5
3 EFT and the new physics regime 8
3.1 Parametrization of the new physics regime 83.2 EFT of the new physics regime and a modified dispersion relation 93.3 The strong coupling scale 11
4 The new physics regime from heavy fields 15
4.1 Integration of a single massive field 154.2 On the role of higher-order time derivatives 17
5 Implications for inflation 19
5.1 Power spectrum 195.2 Bispectrum 21
6 Conclusions 22
A Integration of several massive fields 24
1 Introduction & summary
Effective field theory (EFT) constitutes a powerful and unified scheme to study the possibleeffects of unknown ultraviolet (UV) degrees of freedom on the low-energy evolution of curva-ture perturbations during inflation [1, 2]. By employing general symmetry arguments, it ispossible to deduce the most general EFT parametrizing the low-energy dynamics of curva-ture perturbations on a Friedmann-Lemaıtre-Robertson-Walker (FLRW) background. Thebenefit of adopting such a perspective is manyfold: Firstly, it allows one to focus the studyof inflation on the dynamics of curvature fluctuations — which are ultimately responsible forany low-energy observable today — relegating the model-dependent background dynamicsto a subsidiary plane. Secondly, it offers a simple and intuitive interpretation of curvatureperturbations as Goldstone boson modes, emerging as a consequence of the spontaneousbreaking of time translation invariance. This, in turn, allows the application of well knowntechniques, both perturbative and non-perturbative, to analyze various cosmological observ-ables, such as n-point correlation functions. Thirdly, it offers an explicit parametrization ofall the relevant couplings in the low-energy evolution of curvature perturbations, simplifyingthe study of the relations between field-operators appearing at different orders in perturba-tion theory. All of these characteristics have invigorated the study of inflation, allowing fora model-independent approach to the analysis of the infrared (IR) inflationary observablesaccessible today [3–17].
In the particular case where the inflationary perturbations are generated by a singlescalar degree of freedom, a fully satisfactory effective field theory of their dynamics was firstderived by Cheung et al. in ref. [1]. The basic procedure adopted there was simple and intu-itive: first, one postulates as a background an FLRW geometry that breaks time translation
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invariance but leaves the spatial diffeomorphisms untouched. Then, the spontaneous break-ing of time translation invariance necessarily results in an extra scalar degree of freedomwhich reveals itself as a longitudinal polarization of the metric. In unitary gauge, this scalardegree of freedom may be identified with the 00-component of the metric δg00 ≡ 1+ g00 and,to lowest order in spacetime differential operators, its effective action takes the form
SEFT =
∫
d3xdt√−g
[
M2Pl
2R+M2
PlHg00 −M2Pl(3H
2 + H)
+1
2!M4
2 (t)(g00 + 1)2 +
1
3!M4
3 (t)(g00 + 1)3 + · · ·
−M31 (t)
2(g00 + 1)δKµ
µ − M22 (t)
2δKµ
µ2 − M2
3 (t)
2δKµ
νδKνµ + · · ·
]
, (1.1)
where MPl stands for the Planck mass, H = a/a is the expansion rate defined in terms of thescale factor a(t), and δKµ
ν denotes the perturbed extrinsic curvature of spatial foliations ata fixed time. In the previous expression, the second and third terms are required so that thebackground Friedmann equations are satisfied, independently of the fluid driving inflation.TheMn(t) and Mn(t) coefficients are functions of time only, and parametrize the effects on thelow-energy dynamics due to the unknown UV physics. They should therefore be regarded asundetermined parameters of the theory. For instance, settingMn(t) = Mn(t) = 0 correspondsto the standard case derived from an action for a single canonical scalar field theory with aflat potential [18–20]. It should be clear that the action (1.1) does not reveal the energy scaleΛUV at which ultraviolet degrees of freedom start playing a relevant role in the inflationarydynamics. Nevertheless, one typically expects the coefficients Mn(t) and Mn(t) to dependon a combination of scales involving ΛUV, MPl and the background quantities H and H.
The Goldstone boson field π(x) may be introduced as the adiabatic field fluctuationalong the time direction, in such a way that the spontaneously broken time diffeomorphismt → t+ξ0 is non-linearly realized through the complementary field transformation π → π−ξ0
(see section 2 for more details). At lowest order in perturbation theory, the Goldstone bosonis simply related to the co-moving curvature perturbation ζ(x) by π(x) = −ζ(x)/H. In termsof the Goldstone boson, the effects due toM4
2 are already relevant at the free field theory level,where its appearance results in a reduction of the speed of sound cs at which fluctuationspropagate. Concretely, one finds that the propagation of π-modes is characterized by adispersion relation given by
ω(p) = csp, with1
c2s= 1 +
2M42
M2Pl|H|
,
where p is the momentum carried by the Goldstone boson quanta. Crucially, because 1+ g00
depends non-linearly on π, a suppression of the speed of sound inevitably implies the existenceof higher-order interactions with strengths proportional to M4
2 = (c−2s − 1)M2
Pl|H|/2. As aconsequence, a suppression of the speed of sound at the free field theory level also implies theappearance of non-trivial cubic interactions leading to potentially large levels of equilateral
non-Gaussianity characterized by an f(eq)NL parameter of the form [1, 4]:
f(eq)NL ∼ 1
4c2s. (1.2)
Given that M2 is a mass scale related to the unknown UV physics, unsuppressed by thesymmetries of the background, observation of large non-Gaussianity is quite possible in future
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cosmological probes. Current observational bounds on f(eq)NL imply an approximate lower
bound on cs given by cs > 0.01 [4, 21]. However, the non-linear interactions in action (1.1)induced by cs < 1 are non-unitary and, as such, they imply that the theory becomes stronglycoupled at an energy scale sensitive to cs, given by1
Λ4s.c. = 4πM2
Pl|H|c5s (1− c2s )−1.
Since a reduction in the speed of sound decreases the value of this strong coupling scale,there is a lower bound on how small cs can be or, equivalently, on how large non-Gaussiansignatures are allowed to be without rendering the effective description invalid. A simpleestimation, taking into account the observed value of the amplitude of the primordial powerspectrum Pζ of curvature perturbations, implies the constraint cs > 0.01 in order to avoidstrong coupling of the EFT at energy scales relevant at Hubble horizon crossing2 [1]. Wethus see that this result is consistent with the previous observational constraint.
However, the previous result implies that for small values of the speed of sound (c2s ≪ 1),Goldstone boson modes described by (1.1) may appear strongly coupled at energies well belowthe cutoff energy scale ΛUV at which UV degrees of freedom become excited. This fact reflectsa limitation of the EFT (1.1) to consistently parametrize the class of theories that remainweakly coupled as the energy increases up to ΛUV. In ref. [11], Baumann and Green addressedthis issue by studying weakly coupled completions of (1.1). They pointed out that any fieldtheoretical description remaining weakly coupled all the way up to the symmetry breakingscale will involve either new degrees of freedom or an energy regime with new physics, insuch a way that non-unitary operators stay suppressed. This new physics regime was foundto be characterized by a modified dispersion relation with quadratic momentum dependence
ω(p) ∝ p2,
which can equivalently be seen to lift the strong coupling scale to a value larger than ΛUV, sothat one obtains a low-energy effective description of the system in terms of a weakly coupledGoldstone boson.
The analysis of [11] did not address the more general problem of constructing an EFTfully consistent with the new physics regime, in such a way that the modified dispersionrelation is non-linearly realized to all orders in perturbation theory. The aim of the presentwork is to take a step forward in this direction. Our main emphasis is that the new physicsregime is fully incorporated within the EFT formalism of [1], and a UV-completion is notneeded to keep the theory weakly coupled. This is achieved by allowing action (1.1) toincorporate extra scale-dependent operators acting on the fields. Crucially, we find thatthe action for the Goldstone boson requires a non-trivial modification of its higher-orderinteractions, implying novel effects on the prediction of inflationary observables. Concretely,we show that, in its minimal version, the new physics regime requires a generalization of the
1See appendix E of [11] for a detailed derivation. Note that the expression given in [11] differs by a factorof 4π from that given in [1].
2Although horizon is a widely used term in expressions like ”horizon crossing”, throughout the paper wewill use “Hubble horizon” or “Hubble crossing” instead, to avoid confusion with the particle horizon. See e.g.pg. 40 of [22] for a discussion.
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EFT action (1.1) of the form
SEFT =
∫
d3xdt√−g
M2Pl
2R+M2
PlHg00 −M2Pl(3H
2+H) +M4
2
2!(1+g00)
M2
M2−∇2/a2(1+g00)
+M4
3
3!(1 + g00)
M2
M2 −∇2/a2
[
(1 + g00)M2
M2 −∇2/a2(1 + g00)
]
+ · · ·
, (1.3)
where ∇2 ≡ δij∂i∂j is the Laplacian operator and M is a mass scale parametrizing the newphysics regime. This minimal version is found to be consistent with a single field theorycoupled to a set of heavy fields that are only excited at very high energies (in fact, at anenergy that may be much larger thanM). This may readily be understood by interpreting theoperators (M2−∇2/a2)−1 as propagators characterizing states of massM interacting with theinflaton. Note that this is quite a natural interpretation of the scale-dependent interactions.All UV completions of gravity predict a vast variety of scalar fields (e.g. moduli) that areconstrained to be heavy in order to be compatible with low-energy phenomenology. Sinceone expects that the UV physics is responsible for inflation, logically one also anticipatesthat the inflaton would be coupled to these heavy fields. As we shall see, the new physicsregime appears whenever the Laplacian ∇2 dominates over the mass M in the operator(M2 −∇2/a2)−1, forcing us to consider the formal — but consistent — expansion:
1
M2 − ∇2= − 1
∇2− M2
∇4− · · · , (1.4)
where ∇ ≡ ∇/a. We shall justify this expansion in section 3, where we show that it emerges asa consequence of the non-relativistic nature of the theory, i.e. the breaking of time translationinvariance, which is at the heart of the EFT formulation of inflation. As we shall demonstrate,these modifications imply that the EFT remains weakly coupled all the way up to the energyscale where the integrated UV degrees of freedom become operative, in agreement with theanalysis of ref. [11]. We will argue that these effects are conceivably generic in the sense thatthey consistently capture the gradual way in which UV corrections start to take over untilthe theory may need to be explicitly completed.
Further, we show that the generalized effective action (1.3) implies a non-trivial depen-dence of the inflationary observables on the parameters of the theory. To be more precise, ifthe modes relevant for current observables crossed the Hubble scale within the new physicsregime (that is, if for these modes ω ≃ H when ω ∝ p2) the power spectrum Pζ , the tensorto scalar ratio r and the fNL parameter are found to be given by
Pζ ∼2.7
100
H2
M2Plǫ
√
M
Hcs, r ∼ 7.6ǫ
√
HcsM
, fNL ∼ M
Hcs,
where ǫ = −H/H2 is the usual slow-roll parameter. These expressions differ from thosederived from the standard effective field theory (1.1) and constitute a new parametrizationof the inflationary observables (such as the energy scale of inflation) distinct from othersfound in the recent literature, therefore inviting us to consider more carefully the way inwhich future data should be interpreted.
Although the EFT formalism developed in ref. [1] is entirely general, the bulk of theexisting literature has focussed on analyzing inflation through the specific action (1.1), whichcorresponds to the long wavelength limit of (1.3). In this sense, our work emphasizes the
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strength of the formalism developed in ref. [1] to study inflation more generally, by identifyingnew higher derivative operators that can capture effects of heavy fields coupled to the inflaton,consistent with the EFT formalism [1]. However, we stress that other parametric regimes ofthe EFT formalism, beyond that offered by (1.1), have already been studied in the past (seefor instance refs. [1, 4, 6]).
We have organized this article in the following way: in section 2 we begin by summa-rizing the EFT formalism of ref. [1], focusing on the relation between the Goldstone bosonand co-moving curvature perturbations. In section 3 we present and discuss the basic exten-sion of the effective action to parametrize the new physics regime, including a discussion ofthe strong coupling scale. Then, in section 4, we show how this form of the action emergesfrom the simple case in which a single heavy degree of freedom is integrated out (a moregeneral treatment, where several heavy degrees of freedom are integrated out, is presentedin appendix A). There we also discuss the validity and consistency of the expansion (1.4) toparametrize the effects of UV-degrees of freedom on the low-energy dynamics of curvatureperturbations. In section 5 we discuss the implications of the extended EFT for inflationaryobservables. We show that, indeed, the scales parametrizing the new physics regime modifythe computation of two and three-point correlation functions, implying a re-interpretationof the parametrization of non-Gaussianity in terms of the speed of sound (1.2), or other La-grangian operators [4], which is given by the standard EFT perspective. Finally, in section 6we present our concluding remarks.
2 Review of the standard EFT formalism
Here we review the EFT formalism developed in [1], written in terms of the Goldstone bosonπ and its relation, via gauge transformations, to the co-moving curvature perturbation ζ.Readers familiar with this material may skip this discussion and go directly to section 3.
Our starting point is to consider a quasi de Sitter time-dependent background thatbreaks time diffeomorphisms but keeps spatial ones. To this end, it is useful to adopt theArnowitt-Deser-Misner (ADM) decomposition of the four-dimensional spacetime [44] throughthe metric
ds2 = −N2dt2 + γij(Nidt+ dxi)(N jdt+ dxj), (2.1)
where N and N i are the lapse and shift functions — to be treated as Lagrange multipliers— and γij is the induced metric on the 3-D spatial foliations. In terms of these quantities,the components of the metric gµν and its inverse gµν are given by
g00 = −N2 + γijNiN j , g0i = γijN
j , gij = γij ,
g00 = − 1
N2, g0i =
N i
N2gij = γij − N iN j
N2, (2.2)
where γij is the inverse of γij . Throughout this article we focus our attention on the casewhere primordial perturbations are due to a single scalar degree of freedom. In order tocompute observables such as correlation functions, it is convenient to choose a gauge. Thereare two gauges that are particularly useful for this goal: one is the co-moving or unitary gauge,where the scalar degree of freedom corresponds to the co-moving curvature perturbationζ(x, t) parametrizing inhomogeneous fluctuations of the flat spatial metric as:
gij = a2(t)e2ζδij . (2.3)
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JCAP04(2013)004
If we were considering a model of inflation with a scalar field φ playing the role of the inflaton,in this gauge we would have that its fluctuations are set to zero δφ = 0. The other gauge weconsider is the flat gauge, where the spatial metric corresponds to pure background:
gij = a2(t)δij . (2.4)
In this case, the scalar degree of freedom appears explicitly in the matter sector of the theory,which in the present case does not need to be specified. The transformation between the twogauges is a spacetime re-parametrization [23]. Indeed, in order to go from (2.4) to (2.3) oneperforms a time re-parametrization of the form:
t → t = t+ π(t, x). (2.5)
This introduces the Goldstone boson field π(t, x) as a mean of parametrizing scalar per-turbations in the flat gauge (2.4). The scale factor a(t) changes as a(t) = a(t − π(t, x))which results in a Weyl rescaling a(t) → a(t)eζ(t,x). Thus, to first order in π, one hasa(t) = a(t− π) = a(t)e−Hπ, from which one can deduce the relation ζ = −Hπ. For the com-putation to second and higher order one has to iterate the Taylor expansion and computea(t) = a[t− π(t− π(t, x), x)] and so on. For example the result to second order is
a[t− π(t− π(t, x), x)] = a(t)− a(t)π(t− π(t, x), x) +1
2a(t)[π(t− π(t, x), x)]2 , (2.6)
which gives [3]
ζ(2)(π) = −Hπ +Hππ +1
2Hπ2, (2.7)
where we used π(t−π) = π(t)+O(π2) and the definition of the Hubble parameterH. Becauseof this time re-parametrization the metric picks up non-diagonal terms starting from secondorder in π. These extra terms can be eliminated by performing a space re-parametrizationwith a parameter that is a series in π starting at second order, an operation which results inmore complicated expressions for ζ(π), involving spatial derivatives [23]. The variable ζ inthe comoving gauge stays constant outside the Hubble horizon; its correlators are the gaugeinvariant observables one wishes to compute. Alternatively, as we will see below, the flatgauge can be very useful from a computational point of view, since there exist physical limitswhere the problem simplifies considerably and one is able to draw powerful qualitative andquantitative conclusions.
As stated in the Introduction, the effective field theory describing a single scalar degreeof freedom in the unitary gauge, where there are only metric fluctuations, takes the form
SEFT =
∫
d3xdt√−g
[
M2Pl
2R+M2
PlHg00 −M2Pl(3H
2 + H)
+1
2!M4
2 (t)(1 + g00)2 +1
3!M4
3 (t)(1 + g00)3 + · · ·]
, (2.8)
where we have ignored contributions coming from perturbations of the extrinsic curva-ture δKµ
ν . In order to go to the flat gauge one can invert the aforementioned time re-parametrization (2.5) which may be thought of as the Stuckelberg procedure. This cor-responds to performing a time re-parametrization t → t = t − π(x, t) and assigning thetransformation law π(x, t + ξ(t)) = π(x, t) − ξ(t), such that the combination t + π(x, t) is
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JCAP04(2013)004
invariant under t → t + ξ(t). Thought of in this way one can identify the π field as theGoldstone mode arising from the spontaneous breaking of time diffeomorphisms. Under thistransformation the time component of the metric changes as
g00 =∂x0
∂x′µ∂x0
∂x′νg′µν = (1 + π)2g′00 + 2(1 + π)∂iπg
′0i + g′ij∂iπ∂jπ
= − 1
N2(1 + π)2 + 2(1 + π)
N i
N2∂iπ + a2(t)δij∂iπ∂jπ, (2.9)
where the new metric g′µν is the flat metric (2.4) conveniently expressed in the ADMparametrization (2.2). Therefore, the action (2.8) in the flat gauge takes the form:
S[π] =
∫
dx3dt√−g
[
1
2M2
PlR−M2Pl(3H
2(t+ π) + H(t+ π))
+M2PlH(t+ π)
(
− 1
N2(1 + π)2 + 2(1 + π)
N i
N2∂iπ + gij∂iπ∂jπ
)
+∞∑
n=1
M4n(t+ π)
n!
(
− 1
N2(1 + π)2 + 2(1 + π)
N i
N2∂iπ + gij∂iπ∂jπ + 1
)n]
.
(2.10)
Although this is quite a complicated action when expanded out, there is a physical limit onecan take where the whole situation simplifies considerably. This is the so-called decouplinglimit [1], in which the effects of gravity decouple from the scalar degree of freedom. From thesecond line of the action (2.10) one can see that the leading mixing term of the scalar modewith gravity is of the form M2
PlHπδg00, where δg00 ≡ 1/N2 − 1. Then, after canonicallynormalizing the fields using πc = MPl|H|1/2π/cs, and δg00c = MPlδg
00, this term reads√ǫHcsπcδg
00c . Thus one may conclude that for energies
ω ≫√ǫHcs, (2.11)
one can neglect such mixing terms and consider only the dynamics of the Goldstone mode.This is the analogue of the equivalence theorem [24, 25] which states that in the context ofa spontaneously broken gauge symmetry, there exists an energy above which the would-beGoldstone mode decouples from the gauge field, and becomes a dynamical scalar degree offreedom. As we see from (2.11), this limit corresponds to the slow-roll approximation inthe inflationary context. This fact can also be nicely demonstrated by writing the quadraticaction to first order in slow-roll.
One can compute the action S[π] to any desired order in π. For instance, using theADM decomposition explicitly with N ≡ 1 + δN , we can rewrite the quadratic part of theaction (2.10) in the decoupling limit as
S(2) = M2Pl
∫
d3xdta3[
− 3H2δN2 − 6HHπδN − 3H2π2
−2∂iNi(HδN + πH) + H
(∂π)2
a2− H
c2s(δN − π)2
]
, (2.12)
where we have defined the speed of sound cs through the relation
1
c2s= 1 +
2M42
M2Pl|H|
.
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The linear constraint equations may be solved to give
δN = ǫHπ, ∂iNi = − ǫ
c2s
d
dt(Hπ). (2.13)
After substituting these relations back into the action, we obtain the second-order action
S(2) = −M2Pl
∫
d3xdta3H
[
(ǫHπ − π)1
c2s(ǫHπ − π)− (∂π)2
a2
]
. (2.14)
Finally, from (2.7) truncated to first order so that ζ = −Hπ, we immediately see that
S(2) = M2Pl
∫
d3xdta3ǫ
[
ζ2
c2s− (∂ζ)2
a2
]
. (2.15)
The action (2.14) contains a mass term for π, coming from mixing with gravity, whose form forthe canonically normalized field πc is ǫH
2π2c . Therefore, the deeper one goes in the decoupling
limit, the closer to massless the π field becomes, approaching a true Goldstone mode. Sincethis mass is of order the slow-roll parameters, the decoupling limit coincides with a slow-rollexpansion. In this limit, the flat gauge and particularly the EFT formalism in this gauge,can be the most useful description of inflationary perturbations.
3 EFT and the new physics regime
In this section, we reformulate the standard EFT action (1.1) such that it includes a regimecompatible with modified dispersion relations, which we derive, showing explicitly that thislifts the strong coupling scale well above the UV cutoff scale ΛUV. This gives a generalizedformalism which is weakly coupled throughout the low-energy regime and describes newphysics in the form of a modified dispersion relation and scale-dependent interactions.
3.1 Parametrization of the new physics regime
Here we show that action (1.1) has a natural extension in which theM4n(1+g00)n/n! terms are
modified in such a way that the EFT parametrizes the new physics regime. In what followswe drop the extrinsic curvature terms by setting Mn = 0 and focus on the scalar sector ofthe theory. As a concrete guideline, we start by examining the extension to (1.1) due tothe presence of heavy fields interacting with the gravitational potential (1 + g00). The mainidea is the following: terms beyond the single scalar field paradigm (Mn = 0) will appearas effective interaction terms resulting from the mediation of massive particle states withpropagators proportional to (M2 − 2)−1, where M is the mass of the field being integratedout. In this context, we expect that contributions to the EFT due to these propagators ingeneral imply the following effective n-point interaction term:3
L(n)EFT ∝
[
(g00 + 1)M2
M2 −2
]n−1
(g00 + 1). (3.1)
At first, the extension implied by such terms seems irrelevant. Indeed, at low energies onewould expect that any contribution coming from 2 ≡ −∂2
t + ∇2 acting on (g00 + 1) willremain suppressed with respect to the mass scale M2, hence justifying the usual expansion:
1
M2 −2=
1
M2+
2
M4+ · · · . (3.2)
3For simplicity, we momentarily disregard gravity, allowing ourselves to drop any effects coming from thetime variation of the scale factor a(t). We will amend this omission immediately, in the analysis of section 3.2.
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JCAP04(2013)004
This in turn implies that L(n)EFT ∝ (g00 + 1)n constitutes the leading order contribution to
the effective action induced by heavy fields, bringing us back to the standard result (1.1).However, because time translation invariance is broken, the system may find itself in anon-relativistic regime where the expansion (3.2) becomes a poor representation of the low-energy kinematics. In a non-relativistic time-dependent background, we are instead allowedto consider the case in which a hierarchy of scales appears between frequency and momenta,leading to the more general possibility:
1
M2 −2=
1
M2 − ∇2− ∂2
t
(M2 − ∇2)2+ · · · . (3.3)
It is clear that such an expansion is only possible in a regime where the propagation ofGoldstone bosons is characterized by a non-relativistic dispersion relation ω(p) satisfying
ω2 ≪ M2 + p2, (3.4)
which in turn defines the low-energy regime. Just like in (3.2), the expansion in (3.3) impliesthat additional degrees of freedom other than the Goldstone boson — due to the higher timederivatives — will remain non-dynamical at low energies. In particular, the splitting impliedby (3.4) allows for physical situations where the momentum is much larger than the massscale (i.e. p2 ≫ M2) without the appearance of additional UV degrees of freedom. The endresult is a low-energy effective field theory with scale-dependent n-point interactions of thegeneral form:
L(n)EFT ∝
[
(1 + g00)M2
M2 − ∇2
]n−1
(1 + g00). (3.5)
In what follows, we analyze the direct consequence of having a low-energy EFT with contri-butions of the form given by (3.5). In section 4 we present a concrete realization where (3.5)is obtained from a well-defined system where the inflaton interacts with a single massive field.There, we also show that the massive field leading to (3.5) becomes a Lagrange multiplier atlow energies.
3.2 EFT of the new physics regime and a modified dispersion relation
Following the previous discussion, we now consider the full EFT action, taking into accountthe n-point contributions of the form (3.5) and including gravity. This leads to:
SEFT =
∫
d3xdt√−g
[
M2Pl
2R+M2
PlHg00 −M2Pl(3H
2 + H)
+∞∑
n=2
M4n
n!
[
(1 + g00)M2
M2 − ∇2
]n−1
(1 + g00) + · · ·]
. (3.6)
We shall justify this form in section 4, where we deduce it in the specific example where theGoldstone boson is coupled to a single heavy field by turns of the inflationary trajectory infield space. A more general argument is given in appendix A. Notice that we have constructedthis action by inserting the ratio M2/(M2 − ∇2) to keep the Mn parameters dimensionful.In this way, the resulting EFT theory will be completely parametrized by the M4
n coefficientsand the new mass scale determined by M . As discussed in section 2, the Goldstone boson isrelated to g00 of the unitary gauge by
g00 = − 1
N2(1 + π)2 + 2(1 + π)
N i
N2∂iπ + a2(t)δij∂iπ∂jπ, (3.7)
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JCAP04(2013)004
where N and N i are the lapse and shift functions of the ADM decomposition (2.1) and a(t)is the scale factor of the FLRW background. Then, by writing the action in terms of theGoldstone boson mode π up to cubic order following the same steps shown in section 2,we obtain
S(3) = −M2Pl
∫
d3xdta3H
[
π
(
1 +2M4
2
M2Pl|H|
M2
M2 − ∇2
)
π − (∇π)2]
+
∫
d3xdta3[
2M42
(
π2 − (∇π)2) M2
M2 − ∇2π − 4
3M4
3
(
πM2
M2 − ∇2
)2
π
]
. (3.8)
Notice that the standard EFT action for the Goldstone boson is recovered by taking theformal limit M2 → ∞. In the present case, the Goldstone boson has acquired a non-trivialkinetic term with a strong scale dependence. As a consequence, the dispersion relationcharacterizing the free theory is
ω2(p) =M2 + p2
M2c−2s + p2
p2, (3.9)
where cs is the speed of sound defined in the long wavelength limit as
1
c2s= 1 +
2M42
M2Pl|H|
, (3.10)
and p ≡ k/a is the physical momentum. Recall that the expansion in (3.3) is valid only inthe low-energy regime defined by ω2 ≪ M2 + p2. In terms of momentum, this condition isequivalent to
p2 ≪ M2c−2s . (3.11)
In this limit the dispersion relation may be expanded as
ω2(p) = c2sp2 +
(1− c2s )
M2c−2s
p4 +O(p6), (3.12)
the term proportional to p6 being always subleading. The cutoff energy ΛUV determining thevalidity of this expansion can be estimated by evaluating ω at the value p = Mc−1
s , giving
Λ2UV ∼ M2c−2
s . (3.13)
Thus, ΛUV represents a simultaneous cutoff scale for both momentum p and energy ω. Abovethis scale, the expansion (3.3) breaks down explicitly and the system has to be UV-completedin such a way that it incorporates the states characterized by the mass scale Mc−1
s as newdegrees of freedom. The value of c2s determines the size of the non-trivial effects due to thepropagators coming from the heavy physics sector. The quadratic piece in (3.12) dominateswhen p2 ≪ M2, whereas the quartic term dominates when the momentum is in the rangeM2 ≪ p2 ≪ M2c−2
s . The associated threshold energy scale is Λnew ∼ Mcs, found byevaluating ω2 at p2 ∼ M2. We thus see that interesting effects related to the running ofmomentum in the EFT coefficients only occur for c2s ≪ 1. We may rewrite the new physicsaction in a way that incorporates cs explicitly as
S = −M2Pl
∫
d3xdta3H
[
π(
1 + Σ(∇2))
π − (∇π)2 +[
π2 − (∇π)2]
Σ(∇2)π
− 2M43 c
2s
3M42 (1− c2s )
πΣ(∇2)(
πΣ(∇2)π)
]
, (3.14)
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JCAP04(2013)004
where Σ(∇2) is a differential operator given by
Σ(∇2) = (1− c2s )M2c−2
s
M2 − ∇2. (3.15)
We now see that Σ(∇2) determines the structure of interactions at both quadratic and cubicorder. As the energy increases, the scale dependence of Σ(∇2) affects the strength of selfinteractions of π, potentially modifying the computation of n-point correlation functions, andany phenomenological conclusions derived from it. In particular, provided that c2s ≪ 1, inthe new physics regime one has
Σ(∇2) → −M2c−2s
∇2. (3.16)
Finally let us clarify that, just as for the case described by the effective theory (1.1), itsextended version (3.8), taken on its own, provides no explicit information about the value ofthe UV cutoff scale ΛUV at which the effective field theory breaks down. In other words, thetheory (3.8) may be taken literally as it reads all the way up to momenta p ≫ Mc−1
s , forwhich the dispersion relation (3.9) becomes ω2(p) ≃ p2, consistent with a Lorentz invariantspectrum of massless particles. However, to keep our discussion on firm physical grounds,we assume a UV cutoff ΛUV, consistent with there being a regime of UV physics where theGoldstone boson interacts with one or more heavy fields.
3.3 The strong coupling scale
We now deduce an immediate consequence of the scale-dependence of Σ(∇2): we derive theenergy scale Λs.c. at which the theory (3.14) becomes strongly coupled. Before proceeding,let us briefly summarize the scales that appear in the problem. These are:
• Λnew: the new physics scale which signals the change from a linear to a non-lineardispersion relation.
• ΛUV: the UV cutoff scale given in (3.13), above which all scalar fields are dynamicaland the single field effective description is no longer applicable.
• Λs.c.: the strong coupling scale at which a perturbative approach becomes inconsistent,resulting in a breakdown of the effective description.
• Λs.b.: the symmetry breaking scale at which time diffeomorphisms break and the effec-tive description on a time-dependent gravitational background becomes available.
So far we have identified that the extended EFT (3.14) implies Λnew ∼ Mcs and ΛUV ∼ Mc−1s .
Then, a suppressed speed of sound automatically induces the hierarchy
Λnew ≪ ΛUV, (3.17)
but it tells us nothing about the relative values of Λs.c. and Λs.b. with respect to ΛUV, astheir values strongly depend on the specific UV realization of the inflationary model at hand.Nevertheless, if the UV physics allowing for the existence of a new physics energy regime isalso responsible for generating inflation, it is reasonable to expect all of these scales to be ofthe same order:
ΛUV ∼ Λs.b. ∼ Λs.c.. (3.18)
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JCAP04(2013)004
We emphasize that the present analysis is strictly valid only for inflationary models with asingle scalar degree of freedom driving inflation, and that models with multiple degrees offreedom will inevitably introduce a larger set of scales into the problem.
First, following the discussion of ref. [11], it is possible to deduce that the symmetrybreaking scale Λs.b., taking into account the fact that the dispersion relation scales as ω ∼ p2,is given by:
Λs.b. =
[
2M2Pl|H|
Λ4UV
]2/7
ΛUV. (3.19)
This result allows us to see that the value of Λs.b. compared to ΛUV depends on the ratio2M2
Pl|H|/Λ4UV. For instance, if the UV physics in charge of modifying the low-energy dy-
namics of curvature perturbations is also responsible for producing inflation, it is perfectlyfeasible to have Λ4
UV ∼ 2M2Pl|H|, implying Λs.b. ∼ ΛUV. For the benefit of the present dis-
cussion, we adopt the optimistic perspective whereby Λs.b. ∼ ΛUV, consistent with a largevalue for the slow-roll parameter ǫ compatible with observations (see section 5).
We now proceed to calculate the strong coupling scale Λs.c., that is, the scale at whichtree-level interactions violate unitarity. As discussed in ref. [1], a reduced speed of soundc2s < 1 inevitably introduces non-unitary self-interactions for the Goldstone boson that makethe theory (1.1) strongly coupled at an energy scale Λs.c. given by:
Λ4s.c. = 4πM2
Pl|H|c5s (1− c2s )−1. (3.20)
However, this result is strictly valid only for the case in which ω2 = c2sp2, characteristic
of the standard EFT picture. In fact, in ref. [11] it was found that a modification of thedispersion relation will generally alleviate the strong coupling problem by making Λs.c. largerthan the value of (3.20). Nevertheless, in that work a general analysis incorporating the scaledependence of self-interactions consistent with the modification of the dispersion relation wasnot taken into account. In what follows we incorporate this aspect into the analysis of strongcoupling and show that the conditions for the theory to remain weakly coupled are satisfiedall the way up to an energy scale of the same order as, or larger than, the natural cutoffΛUV = Mc−1
s of the new physics regime.
To proceed, we will follow the analysis of ref. [11] closely. First, by normalizing the Gold-stone boson as πn = (2M2
PlǫH2)1/2π the quadratic part of the extended EFT action (3.14)
may be written as
S =1
2
∫
d3xdt
[
πn
(
M2c−2s −∇2
M2 −∇2
)
πn − (∇πn)2
]
. (3.21)
Notice that we have fixed a = 1 for the sake of simplicity, assuming that our discussioninvolves processes at energy scales much larger than H. Now, the conjugate momentumPπ ≡ ∂L/∂πn of this free field theory is given by
Pπ =M2c−2
s −∇2
M2 −∇2πn. (3.22)
This implies that the commutation relation [πn, Pπ] = iδ reads as
[
πn(x1) ,M2c−2
s −∇22
M2 −∇22
πn(x2)
]
= iδ(x1 − x2), (3.23)
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JCAP04(2013)004
where ∇22 stands for a Laplacian operator written in terms of the coordinate x2. Another
way of writing this expression is:
[πn(x1) , πn(x2)] = iM2 −∇2
2
M2c−2s −∇2
2
δ(x1 − x2). (3.24)
Then, to satisfy these commutation relations, we may consider the quantization of the freefield πn(x) in terms of creation and annihilation operators a†(k) and a(k) satisfying[a(k1), a
†(k2)] = δ(k1 − k2). We find
πn(x) =1
(2π)3/2
∫
d3p[
πn(p)a(p)e−iωt+ip·x + πn(p)
∗a†(p)e+iωt−ip·x]
, (3.25)
where πn(p) corresponds to the field amplitude in Fourier space, given by:
πn(p) =
√
M2 + p2
M2c−2s + p2
1√
2ω(p)=
√
ω(p)
2p2. (3.26)
Notice that due to the modified commutation relation (3.23), the functional form of πn(p)differs substantially from the standard Minkowskian result 1/
√2ω. Nevertheless, it is possible
to see that when p ≪ Λnew = Mcs we recover back the standard amplitude 1/√2csp after
canonically normalizing πc = πn/cs. Apart from this modification, the quantization of thequadratic action proceeds in the usual way.
Let us now move on to consider the interacting part of the theory. Notice that therelevant quartic interaction due to M4
2 , coming from the non-linear self-interactions in theEFT is given by:4
Lint =(1− c2s )
16M2PlǫH
2(∇πn)
2 M2c−2s
M2 −∇2(∇πn)
2, (3.27)
after taking into account the normalization πn = (2M2PlǫH
2)1/2π. Then, we can analyze theeffect of this interaction on the tree-level scattering of two π fields into two final π’s in thecenter of mass reference frame. The main point to be kept in mind when performing thiscomputation is that the new amplitude (3.26) for the quantized Goldstone boson field impliesthat each external leg of the relevant diagram will come with an additional factor
√
M2 + p2iM2c−2
s + p2i, (3.28)
where pi is the momentum carried by the particle represented by the ith external leg of thediagram. After a straightforward computation, we find that the scattering amplitude of thisinteraction is given by:
A(p1, p2 → p3, p4) =(1− c2s )c
−2s p4
2M2Pl|H|
(
M2 + p2
M2c−2s + p2
)2
×[
1 +M2 cos2 θ
M2 + 2p2(1− cos θ)+
M2 cos2 θ
M2 + 2p2(1 + cos θ)
]
, (3.29)
4Another relevant interaction that could contribute to this analysis is the one proportional to Lint ∝
π2Σ(∇2)π2. However, in the new physics regime one has ω
4≪ p
4, implying that this interaction will besubstantially suppressed compared to (3.27).
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JCAP04(2013)004
where θ is the angle of scattered particles with respect to the impact axis. By recalling that(M2 − ∇2)−1 can be interpreted as the propagator of a heavy field, the first, second andthird terms in the square bracket of (3.29) may be thought as contributions coming fromthe interchange of a heavy boson through the s, t and u channels respectively. The previousresult may be expressed by the partial wave expansion
A(p1, p2 → p3, p4) = 16π
(
∂ω
∂p
ω2
p2
)
∑
ℓ
(2ℓ+ 1)Pℓ(cos θ)aℓ, (3.30)
where the Pℓ(cos θ) are Legendre polynomials. Then the lowest order coefficient a0 is given by:
a0 =(1− c2s )c
−2s p4
32πM2Pl|H|
(
∂p
∂ω
p2
ω2
)(
M2 + p2
M2c−2s + p2
)2 ∫ 1
−1d cos θ
(
1 +2(M2 + 2p2)M2 cos2 θ
(M2 + 2p2)2 − 4p4 cos2 θ
)
=(1− c2s )c
−2s p4
16πM2Pl|H|
(
∂p
∂ω
p2
ω2
)(
M2 + p2
M2c−2s + p2
)2
×[
1 +M2(M2 + 2p2)
2p4
(
M2 + 2p2
4p2log(
1 +4p2
M2
)
− 1
)]
. (3.31)
In order to preserve the unitarity of the tree-level scattering process under consideration,the optical theorem leads to the constraint aℓ + a∗ℓ 6 1. Our main interest is to assess theunitarity of the EFT above Λnew, where the dispersion relation has the form:
ω2 ≃ p4
M2c−2s
. (3.32)
In this regime, the second term in the square bracket of (3.31) becomes negligible, leading tothe result
a0 ≃(Mc−1
s )3/2ω5/2
32πM2Pl|H|c2s
. (3.33)
Then, using the constraint aℓ + a∗ℓ 6 1, i.e. Re(aℓ) <12 , for the particular case ℓ = 0, we find
that the theory remains weakly coupled as long as
ω5/2 < 8πc2s
[
Λs.b.
ΛUV
]7/2
Λ5/2UV. (3.34)
From this result we deduce that the strong coupling scale is given by:
Λs.c. = (8πc2s )2/5
[
Λs.b.
ΛUV
]7/5
ΛUV, (3.35)
where Λs.b. is the symmetry breaking scale (3.19). Equation (3.35) admits a variety ofsituations depending on the values of the scales Λnew, ΛUV and Λs.b.. For instance, if wetake Λs.b. ∼ ΛUV and c2s = Λnew/ΛUV ∼ 10−4, then (8πc2s )
2/5 ∼ 0.1 and the value of Λs.c.
is found to be of order ΛUV. However, if Λs.b. ≫ ΛUV, then one can have models withc2s ≪ 10−4 and still satisfy Λs.c. ∼ ΛUV. Thus we see that the non-trivial modificationscharacterizing the new physics regime M2 ≪ p2 ≪ M2c−2
s imply that the interactions ofthe theory scale differently with energy, changing significantly the value at which the EFTbecomes strongly coupled.
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JCAP04(2013)004
4 The new physics regime from heavy fields
Actions containing an arbitrary number of inverse differential operators, such as (3.6), areknown to be non-local, potentially suffering from classical instabilities and the appearanceof ghosts at the quantum level, when the non-local terms can be written as the limit of asequence of higher-derivative terms. Indeed, Ostrogradski found that theories which dependnon-trivially on more than one time derivative (i.e. in such a way that the higher derivativescannot be removed by integrating by parts) are unstable, with their Hamiltonians unboundedfrom below [26, 27]. Upon quantization the instability persists, manifested by the appearanceof negative norm states or ghosts, which in turn translates into loss of unitarity. However,when the theory in question corresponds to an effective field theory derived from a localtheory by integrating out one or more fundamental dynamical variables, we do not encounterthese problems. In such a case, because the original theory is local, it is not valid to considerthe resulting non-local terms as limits of higher-derivative terms [28, 29], implying that thereare no problems either with instabilities or ghosts, as long as the theory remains within itsdomain of validity.
In what follows we explicitly relate the non-local form of action (3.6) to the presenceof additional degrees of freedom that become operative at high energies. We show that thetheory at hand becomes ill defined only if one insists in assuming its validity at energies oforder Mc−1
s , where a second degree of freedom inevitably becomes excited. The result is thatat low energies the theory (3.6) is safe from any pathology related to non-locality.
4.1 Integration of a single massive field
As an illustrative example, let us study the specific case of the EFT obtained by inte-grating out a single massive field parametrizing deviations from the trajectory in fieldspace [16, 30–33].5 This gives rise to terms with insertions of the form M2
M2−∇2 , providinga physical motivation for the modified action (1.3). In the more general case, one obtains
a more complicated function of the momenta and couplings instead of the insertion M2
M2−∇2 ,and a more complicated dispersion relation (see appendix A for the relevant analysis). Inunitary gauge, the quadratic action for a single heavy field fluctuation F coupled to theinflaton, is given by
SF =1
2
∫
d3xdta3
F2 − (∇F)2 −M2F2 − 2θφ0F(g00 + 1)− θ2F2(g00 + 1)
, (4.1)
where θ is the angular velocity characterizing the turns of the multi-field trajectory in thescalar field target space, and φ0 is the rapidity of the background scalar field. Then, theequation of motion of the heavy field reads
F + 3HF +(
M2 −∇2 + θ2(g00 + 1))
F = −θφ0(g00 + 1). (4.2)
We are interested in studying the low-energy regime of the system, where the second-ordertime variation of the heavy field F is subleading with respect to the term (M2 −∇2)F . Asshown in ref. [32], in order to integrate out F it is also important to assume the adiabaticitycondition |θ/θ| ≪ M , which ensures that the background dynamics of the turning trajectory
5The potentially large influence that heavy fields could have on the low-energy dynamics of inflation wasfirst emphasized by Tolley and Wyman in ref. [34]. For other recent approaches studying the effects of heavyfields on the low-energy dynamics of inflation, see for instance refs. [35–43].
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JCAP04(2013)004
are consistent with the condition ω2 ≪ M2 + p2. If this is granted, we may simply disregardthe kinetic term and solve for F by rewriting the equation of motion (4.2) as
[
1 + (g00 + 1)θ2
M2 −∇2
]
(M2 −∇2)F = −θφ0(g00 + 1). (4.3)
This implies that the heavy field F may be treated as a Lagrange multiplier, allowing us toexplicitly write it in terms of g00 + 1 as
F = − θφ0
M2 −∇2 + θ2(g00 + 1)(g00 + 1)
= −θφ01
M2 −∇2
[
1 + (g00 + 1)θ2
M2 −∇2
]−1
(g00 + 1)
= −θφ01
M2 −∇2
∞∑
n=0
(−1)n
[
(g00 + 1)θ2
M2 −∇2
]n
(g00 + 1), (4.4)
where in the last step we made use of the formal expansion (1 + x)−1 =∑
n(−1)nxn, validfor |x| < 1. Notice that neglecting the kinetic term at the level of the equations of motion isequivalent to having dropped them in the action. Then, disregarding the kinetic term for Fin the action and inserting (4.4), we recover the contribution to the EFT for the Goldstoneboson coming from the heavy field:
SF = −M2Pl
∫
d3xdta3H
∞∑
n=2
(−1)n[
(g00 + 1)θ2
M2 −∇2
]n−1
(g00 + 1), (4.5)
where we have used the background equation φ20 = −2HM2
Pl. Assuming that there are noadditional sources of deviations from standard single field inflation other than the heavy fieldF , it is straightforward to deduce that the speed of sound is given by
1
c2s= 1 +
4θ2
M2, (4.6)
implying that the M4n coefficients of the EFT action (1.1) may be written as [16]
M4n = (−1)n|H|M2
Pln!
(
1− c2s4c2s
)n−1
. (4.7)
Thus, we have obtained the new physics EFT action (3.6) by integrating out a single heavyfield. In this case the parameters acquire a specific dependence on the background quantity θwhich parametrizes the UV-complete theory (4.1). Different parameters are obtained in thegeneral case, where one may have coupling to more than one heavy field (see appendix A).It is important to recognize that at low energies the massive scalar field F has no dynamics,in the sense that its value is completely determined by the source −θφ0(g
00 +1) at the righthand side of (4.3). In other words, the heavy field F plays the role of a Lagrange multiplier,carrying with it the scale dependence implied by the ∇2 operator.
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JCAP04(2013)004
4.2 On the role of higher-order time derivatives
We are now in a position to take a closer look at the role played by higher-order timederivatives in the low-energy effective field theory. As already argued, the expansion (3.3) isonly possible if Lorentz invariance is broken, which in the present context is a consequence ofthe broken time translation invariance induced by the background. In Fourier space eq. (3.3)may be expressed as
1
M2 + p2 − ω2=
1
M2 + p2+
ω2
(M2 + p2)2+ · · · . (4.8)
Here we wish to address the relevance of ω2 in this expansion, and its effect on the low-energydynamics. To proceed, let us consider the quadratic action for π obtained by integrating outthe heavy field F of the previous section, but this time keeping the time derivative ∂t to allorders. Then, the effective action obtained for π is found to be
S(2)π = −M2
Pl
∫
d3xdta3H
[
π
(
1 +4θ2
M2 −2
)
π − (∇π)2
]
. (4.9)
From this expression, it is possible to read off the propagator D(p2) of the low-energy Gold-stone boson in momentum space, which is found to be given by:
D(p2) ∝ 1
Γ(p2), Γ(p2) = p2 − ω2 − 4θ2ω2
M2 + p2 − ω2. (4.10)
This propagator has two poles, at values ω2+ and ω2
− determined by the following expression:
ω2± =
M2
2c2s+ p2 ± M2
2c2s
√
1 +4p2(1− c2s )
M2c−2s
. (4.11)
A particle state characterized by a propagator with two or more poles is condemned to includeghosts in its spectrum [45, 46], in close connection with our discussion on non-locality at thebeginning of this section.6 However, if we restrict the theory to momenta p ≪ Mc−1
s ,corresponding to the low-energy regime, one finds that ω+ and ω− are well approximated by
ω2+(p) = M2c−2
s +O(p2), (4.12)
ω2−(p) = c2sp
2 +(1− c2s )
2
M2c−2s
p4 +O(p6), (4.13)
where O(p2) and O(p6) denote subleading higher-order terms. We thus see that, as long aswe focus on low-energy processes for which p ≪ Mc−1
s and ω ≪ Mc−1s , intermediate particle
states are characterized by well-defined propagators (away from the dangerous pole ω+, whichhas a negative residue) and the effective field theory (4.9) remains ghost free. Moreover wesee that there is a transition scale Λnew within the low-energy regime at which the dispersionrelation ω−(p) changes from linear to non-linear. For cs ≪ 1, this roughly happens at p = M ,for which the first and second terms in (4.13) compete, giving us back
Λnew ≃ Mcs. (4.14)
6Strictly, this is only true for the case of an analytic generatrix, see [46], so that exceptions such as thatin [47] are possible. We thank Neil Barnaby for pointing this out to us.
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JCAP04(2013)004
Interestingly, (4.13) does not coincide with our dispersion relation (3.12) computed with-out taking into account the time derivative ∂t to all orders. The difference between (4.13)and (3.12) is a factor (1 − c2s ) in front of the quartic piece of the expansion. Nevertheless,because this term is only relevant for c2s ≪ 1, we see that the difference between these twoexpressions is only marginal, justifying the approximation by which one drops higher-ordertime derivatives.
To further appreciate the result above, we may consider again the full model (4.1)coupling the scalar mode (g00+1) to the heavy field F , but this time taking into account thedynamics of the heavy field. In the short wavelength regime, where the role of the Hubbleconstant H may be disregarded, the linear equations of motion for both the heavy field Fand the Goldstone boson π are given by
π −∇2π = −2θ
φ0
F , (4.15)
F − ∇2F +M2F = +2φ0θπ. (4.16)
This pair of fields is non-trivially coupled, implying that the solutions correspond to a linearcombination of two modes, hereby denoted + and −, in the form [30, 33]
π = π+eiω+t + π−e
iω−t ,
F = F+eiω+t + F−e
iω−t , (4.17)
where the two frequencies ω− and ω+ are precisely those given by the expressions (4.11).The pairs (π−,F−) and (π+,F+) represent the amplitudes of low and high frequency modesrespectively. Due to the equations of motion (4.15) and (4.16) they satisfy the followingalgebraic relations
F− =2iθφ0ω−
M2 + p2 − ω2−
π− , π+ =1
φ0
2iθω+
ω2+ − p2
F+ . (4.18)
Notice that in the limit θ → 0 we recover the usual situation whereby F− = π+ = 0, andonly the modes π− and F+ contribute to each field. As discussed in detail in [33], at tree-level, integrating out the heavy field corresponds to truncating the heavy mode of frequencyω+. This is equivalent to disregarding π+,F+ and keeping the low frequency modes in thesolution π = π−e
iω−t, and F = F−e
iω−t. Of course, this step is only consistent if there is a
hierarchy of frequencies ω2− ≪ ω2
+, which from (4.11) necessarily implies
p2 ≪ M2c−2s . (4.19)
This corresponds to the low-energy regime, and coincides with our previous criterion foravoiding ghosts at the effective field theory level. This result is twofold: first it showsexplicitly how the appearance of ghosts at the effective field theory level is directly related tothe appearance of heavy degrees of freedom in the full UV-complete theory and, in addition,it provides a physical explanation of the appearance of the scale Λnew. Upon integrating outthe heavy field F , the dynamics of the light field π may be thought as those corresponding tothe propagation of fluctuations in a medium. From (4.13) we see that ω−(p) has the followingform in the new physics regime Λnew ≪ ω− ≪ ΛUV:
ω− ≃ p2
ΛUV+
1
2Λnew. (4.20)
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JCAP04(2013)004
This is a Schrodinger dispersion relation with a mass gap Λnew/2 which corresponds to thescale where particle-like excitations with a non-linear dispersion relation start dominatingover phonon-like ones with a definite speed of sound. Note that the effective mass of theexcitation is set by the UV physics, which is responsible for the lowering of the propagationspeed. Such “gapped” Hamiltonians are familiar from many condensed matter systems, suchas super- or semi-conductors for example.
5 Implications for inflation
We now discuss the observational consequences of the non-trivial modifications emergingfrom the extended EFT of inflation (1.3). We are particularly interested in the distributionof curvature perturbations arising from modes which cross the Hubble scale (ω2 = H2) withinthe new physics regime
M2 ≪ p2 ≪ M2c−2s . (5.1)
Given that in this range ω is dominated by the quadratic piece (ω ∝ p2), we see that in termsof H the regime we are interested in is characterized by the condition
M2c2s ≪ H2 ≪ M2c−2s . (5.2)
As we shall see, the observables computed in this situation do not depend on cs directly —as opposed to the standard case — but instead depend on the combination ΛUV ≡ Mc−1
s .This result changes completely the interpretation of cosmological observations as they relateto theoretical inflationary parameters such as H and ǫ.
5.1 Power spectrum
First, let us consider the derivation of the power spectrum. To proceed, we may consider thequadratic part of the action (3.14) which, for the normalized field πn = (2M2
PlǫH2)1/2π, reads
S =1
2
∫
d3xdta3[
πn
(
M2c−2s − ∇2
M2 − ∇2
)
πn − (∇πn)2
]
, (5.3)
where ∇2 ≡ ∇2/a2. The equation of motion for πn(k) in momentum space is given by
πn + 3Hπn +2H(1− c2s )M
2k2/a2
(M2 + k2/a2)(M2 + c2sk2/a2)
πn + ω2πn = 0, (5.4)
where k corresponds to co-moving momentum, and ω2 is given by eq. (3.9) with p = k/a. Itmay be seen that the third term in (5.4) consists of a non-trivial contribution due to the scale-dependent modification of the kinetic term, whereas the fourth term contains informationabout the dispersion relation ω = ω(p).7 The equation of motion (5.4) cannot be solvedanalytically in full generality due to the time dependence of the coefficients through thescale factor a(t). In order to obtain interesting results we will therefore assume that Hubble
7The extrinsic curvature terms appearing in action (1.1) do imply a modified dispersion relation, but donot generate an equation of motion such as (5.4). The role of these extrinsic curvature terms on inflationaryobservables was studied in detail in refs. [48–50].
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JCAP04(2013)004
crossing takes place in the non-linear dispersion regime and keep the leading terms in theexpansion. Then, using the condition (5.1), the problem simplifies to
πn + 5Hπn +k4
a4Λ2UV
πn = 0. (5.5)
Inspection of this equation allows us to see that Hubble crossing happens at p2 = k2/a2 =HΛUV, consistent with the condition ω = H as long as we take the modified dispersionrelation (3.12). The solution to the equation of motion (5.5) can be expressed in terms of
the Hankel function of the first kind H(1)5/4(x), as [11]
π(τ) =H2τ2
(2M2PlǫH
2)1/2
√
π
8
k
ΛUV
√−τH
(1)5/4(x), x =
H
2ΛUVk2τ2, (5.6)
where τ = −(Ha)−1 is the usual conformal time. To obtain this solution one imposes thecommutation relation (3.23) and chooses initial conditions such that at sub-horizon scalesonly positive frequency modes contribute to the propagating modes. Technically, for thisprocedure to be reliable, we must assume that these initial conditions are imposed within thelow-energy regime ω2 ≪ Λ2
UV. Taking the super-horizon limit of (5.6) and using the relationζ = −Hπ, we then obtain the power spectrum
Pζ =k3
2π2|ζk|2 =
Γ2(5/4)
π3
H2
M2Plǫ
√
ΛUV
H, (5.7)
where Γ(5/4) ≃ 0.91 is the usual Gamma function evaluated at 5/4. As mentioned earlier, thepower spectrum does not depend directly on the value of cs. Because ΛUV ≫ H we see thatthe amplitude of the power spectrum is greatly enhanced (by a factor of cs
√ΛUVH) compared
to the standard result found in single field slow-roll inflation. As a consequence , conclusionsabout the value of ǫ from observations of the power spectrum will change compared to thestandard case. The normalization of the power spectrum implied by WMAP7 is given byPζ = 2.42×10−9 [21], implying the following relation among the parameters determining thepower spectrum
H2
M2Plǫ
√
ΛUV
H∼ 9× 10−8. (5.8)
In terms of the symmetry breaking scale (3.19) introduced earlier, the previous relation reads
Λ2s.b./H
2 ∼ 104. (5.9)
If ΛUV ∼ Λs.b. the previous relation implies a large hierarchy of scales between the Hubblehorizon and the scales involved in the ultraviolet physics. Furthermore, given that we areconsidering only a modification of the scalar sector of the theory, we may safely assume thattensor perturbations remain unmodified by the presumed ultraviolet physics involved in thepresent analysis. This implies that the tensor-to-scalar ratio r predicted for this class ofEFTs has the form:
r =2πǫ
Γ2(5/4)
√
H
ΛUV. (5.10)
Future CMB experiments could constrain the tensor-to-scalar ratio to r < 0.01 [51], implyinga constraint on the parameters of the form ǫ
√
H/ΛUV <∼ 10−3. Then, in the particular casein which ΛUV ∼ Λs.b., we may use (5.9) to deduce ǫ < <∼ 10−2, which would constitute aweaker bound that the one encountered in single field slow-roll inflation (ǫ < 6× 10−4).
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JCAP04(2013)004
5.2 Bispectrum
We now consider the implications of the effective field theory (3.14) on the bispectrum. Themain interactions leading to new effects are due to M4
2 and M43 in the action (1.3), and are
given by
L(3)M2
= −M2Pla
3|H|(∇π)2Σ(∇2)π, (5.11)
L(3)M3
= M2Pla
3|H| 2M43 c
2s
3M42 (1− c2s )
πΣ(∇2)(
πΣ(∇2)π)
, (5.12)
where Σ was defined in eq. (3.15). To obtain an estimate of the size of non-Gaussianity, weproceed by weighting the strength of the quadratic and cubic operators of the theory, evalu-ated at Hubble crossing in the range (5.2). Then, fNL is approximately given by the relation
L(3)
L(2)∼ fNLζ, (5.13)
where the length scales are replaced by their values during Hubble crossing and ζ = −Hπ.Due to the fact that the operators present in the theory consist of spatial derivatives whichdecay rapidly on super-horizon scales, we expect [1] the dominant contribution to non-Gaussianity to be that where all momenta are of similar magnitude, i.e. of the equilat-eral type. Therefore, if Hubble crossing happened within the new physics regime, thenω2 = p4/(Mc−1
s )2 ≃ H2, implying that p2 ≃ MH/cs. This allows us to consider the follow-ing replacements when evaluating the ratio (5.13):
∇2/a2 → Mc−1s H, ∂t → H, Σ(∇2) → Mc−1
s
H. (5.14)
Let us first estimate the contribution of the quadratic piece:
L(2) = a3M2Pl|H|πΣ(∇2)π ≃ a3M2
Pl|H|HΛUVπ2. (5.15)
The cubic contribution coming from the M42 piece is given by
L(3)M2
= −a3M2Pl|H|(∂π)
2
a2Σ(∇2)π ≃ a3M2
Pl|H|Λ2UVπ
2(−Hπ). (5.16)
Finally, the contribution coming from the M43 piece is given by
L(3)M3
= a32M4
3 c2s
3M42 (1− c2s )
M2Pl|H|
[
πΣ(∇2)]2π ≃ a3
2M43 c
2s
3M42 (1− c2s )
M2Pl|H|Λ2
UVπ2(Hπ). (5.17)
Putting the previous results together, we thus see that the generic prediction is
fNL ∼ ΛUV
H, (5.18)
which implies a large non-Gaussian signature. This result shows explicitly that the scaleΛUV enters directly into the computation of low-energy observables such as the level ofnon-Gaussianity. For instance, if ΛUV ∼ Λs.b. then we obtain a sizable estimation of orderfNL ∼ 102.
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JCAP04(2013)004
We have seen how the magnitude of fNL can be large in the new physics regime whereM2 ≪ p2 ≪ M2c−2
s . In addition, although the dominant contribution to the three-pointfunction will still be of equilateral type, we expect that the change in the dispersion relationsources deviations from the equilateral configuration [52–54]. In our approach though, thereis yet another potential source of novel shapes of the three-point functions. This is the scaledependence of the coefficients in the Lagrangian (1.3) in the non-linear dispersion relationregime, a claim which is currently under investigation.
6 Conclusions
Our work emphasizes once more the power of the effective field theory perspective for studyinginflation [1]. As we have seen, action (1.1) represents the lowest order expansion, in termsof spacetime derivatives, of the most general EFT of inflation driven by a single degree offreedom. As such, it is limited in that for a suppressed speed of sound cs ≪ 1 its strongcoupling scale Λs.c. is found to be much lower than the high-energy cutoff scale ΛUV at whichnew degrees of freedom start participating in the inflationary dynamics. This limitation isparticularly critical in the regime where we might optimistically hope that the EFT formalismprovides non-trivial phenomenology through large non-Gaussian signatures. To address thislimitation, in this work we have considered an extension to the standard EFT of inflationmotivated by previously proposed UV-completions, implying a scale-dependent self-couplingfor the Goldstone boson π. This extension involves n-point interactions of the form
L(n)EFT ∝
[
(1 + g00)M2
M2 − ∇2
]n−1
(1 + g00), (6.1)
where 1+ g00 ∼ −2π in the decoupling limit, which we derive by integrating out heavy fieldscoupled to π. A specific example for a single heavy field was presented in section 4.1, whilethe general calculation for arbitrarily many heavy fields is given in appendix A.
The extended EFT implied by (6.1) allows us to access a regime of so-called new physics,characterized by a modified dispersion relation ω(p) which is quadratic in momentum pabove the energy scale Λnew = Mcs, where M is the mass of the field being integratedout. We have shown that, as emphasized in ref. [11], this modified dispersion relation hasimportant consequences for any physical process sensitive to the scaling properties of thevarious operators appearing in the low-energy EFT, including the non-linear self-interactionof the π field. We thus have an EFT description which is valid throughout the low-energyregime, and which is fully consistent with the new physics regime in such a way that themodified dispersion relation is realized by non-linear interactions at all orders in perturbationtheory. This generalizes the analysis of [11] by demonstrating that the new physics regimemay be fully incorporated within the EFT formalism from the very beginning, without theneed for an ad hoc completion to keep the theory weakly coupled. In this particular respect,we have shown explicitly that the scaling properties induced by (6.1) raise the strong couplingscale above the UV cutoff scale ΛUV (section 3.3). This is in part due to the fact that theextended effective field theory (1.3) captures accurately the non-trivial role that UV physicshas on the Goldstone boson as the energy increases, without implying the existence of extradegrees of freedom. In fact the single field EFT description is sensible exactly as far asexpected, that is until the heavy degrees of freedom of the UV complete theory are excited,as we showed in section 4.2.
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JCAP04(2013)004
To summarize our results concerning the phenomenology of inflation, we find that theextended effective field theory (1.3) predicts the following relations between cosmologicalobservables and parameters:
Pζ ≃2.7
100
H2
M2Plǫ
√
ΛUV
H, r ≃ 7.6ǫ
√
H
ΛUV, fNL ∼ ΛUV
H, (6.2)
where ΛUV = Mc−1s . Although the speed of sound cs does not appear explicitly in these
expressions, a suppressed value of it is crucial for the new physics regime to exist. Thesepredictions may be compared to the ones obtained in the case of the standard effective fieldtheory (1.1), given by
Pζ ≃1.3
100
H2
M2Plǫcs
, r ≃ 16ǫcs, fNL ∼ 1
c2s, (6.3)
which are also compatible with the effective field theory (1.3) in the regime ω2 ≪ Λ2new(=
Λ2UVc
4s ). Apart from minor numerical factors, the two sets of predictions differ in their
dependence on the theoretical parameters characterizing the two regimes, such that onemay replace
c2s →H
ΛUV, (6.4)
to go from (6.3) to (6.2). These results suggest that one should be careful when interpret-ing future results from surveys relevant for constraining inflation.8 Adopting an optimisticperspective, an observation of large values for r and fNL in the near future would allow usto infer the values of the parameters ǫ,H, cs in the case we assume the validity of the-ory (1.1) and ǫ,H,ΛUV in the case of theory (1.3). If fNL turns out to be small, it wouldbe impossible to infer the existence of a new physics regime, and we would be forced toconsider the theory (1.1) as the best parametrization of inflation. However, a large value offNL would open up the possibility that Hubble crossing happened within the new physicsregime, implying a drastically different interpretation of the available data. In such a case,it would be preferable to consider a parametrization of inflation consistent with a weaklycoupled description, like the one offered by the extended version (1.3).
It is clear that the examples considered in this work constitute a subset of new physicsmodifications compatible with a consistent EFT formulation. While we expect that any newphysics extension would share similar characteristics to those discussed here, it could be im-portant to re-examine other existing representations motivated by different UV completions,such as DBI-inflation [55]. Even in the present case, there are observational consequences wehave not considered here at all. In terms of the three-point function, many detailed questionsnow arise:
• While we have assumed equilateral type non-Gaussianity and calculated only approxi-
mately the magnitude of f(eq)NL , it is possible that a more careful calculation will reveal
deviations from the equilateral case arising from the modified dispersion relation orsourced by the modified scale-dependent interactions in the theory.
8Recall that other possible parametrizations for these observables, arising from different operators in theLagrangian, are given in [1, 4, 6], for example.
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JCAP04(2013)004
• In estimating f(eq)NL , we also assumed that there were modes which crossed the Hubble
horizon in the range (5.2), and that M2 and M3 were nonzero. It is a non-trivialquestion whether there exist explicit examples, with physically motivated parametersMn, for which this regime will give rise to observable non-Gaussianities.
• It would be interesting to analyze how the present effective field theory could affect othertypes of non-Gaussianity, so far analyzed in the context of the parametrization (1.1),including the so called local- [7] and resonant-type [56, 57].
Last but not least, the modifications studied here could even be relevant to study new physicsfor tensor modes. The effect of the extrinsic curvature terms in (1.1) is limited to a dispersionrelation of the form ω(p) = ch p (where ch = 1 − M2
3 /M2Pl denotes the speed of sound for
tensor-modes), in contrast to the modified dispersion relation they imply for scalars. However,for instance, if tensor modes interacted with additional spin-2 fields during inflation, theirdispersion relation would be non-trivially modified [58], implying an extended EFT withextra spacetime differential operators affecting the extrinsic curvature terms of (1.1).
Acknowledgments
We would like to thank Ana Achucarro, Neil Barnaby, Daniel Baumann, Sebastian Cespedes,Fotis Diakonos, Jinn-Ouk Gong, Subodh Patil and the participants of the “Effective FieldTheory in Inflation” workshop (at the Lorentz Center, Leiden) for useful discussions andcomments on this work. The work of GAP was supported by funds from Conicyt underboth a Fondecyt “Iniciacion” project 11090279 and an “Anillo” project ACT1122. GAPwishes to thank King’s College London, and the University of Cambridge (DAMTP) fortheir hospitality during the preparation of this work. RG was supported by an SFB fellowshipwithin the Collaborative Research Center 676 “Particles, Strings and the Early Universe”and would like to thank the theory groups at DESY and the University of Hamburg for theirhospitality while this work was being carried out. RG is also grateful for the support of theEuropean Research Council via the Starting Grant numbered 256994.
A Integration of several massive fields
Here we justify the form of the interaction terms that appear in the generalized effectiveaction (3.1), upon integrating out several heavy fields. Let us write the simplest actioncoupling multiple heavy fields to δg00 ≡ g00 + 1. We are interested in extracting tree-leveleffects, and therefore we consider an action quadratic in the heavy fields, but to all orders inδg00. To lowest order in δg00, we have
S=−1
2
∫
d3xdt∑
a
Fa
[
−2+M2a−Ba(g
00+1)]
Fa+2Aa(g00+1)Fa+
∑
b
Cab(FaFb)
, (A.1)
where Aa, Bb and Cab are background quantities. The matrix Cab is an anti-symmetricmatrix, 2 corresponds to the FLRW version of the D’Alambertian operator
2 = −∂2t − 3H∂t + ∇2, (A.2)
and the couplings have mass dimensions [A] = 3, [B] = 2, [C] = 1. Notice that we haveexcluded non-diagonal mass terms, which may be eliminated by field redefinitions.
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JCAP04(2013)004
To proceed, we neglect the friction terms coming from the volume factor a3 in d3xdt,and focus on the general structure stemming from integrating out the massive fields Fa. Themore elaborate case in which the friction term is incorporated is completely analogous. Theequations of motion are:
(−2+M2a −Ba(g
00 + 1))Fa +∑
b
CabFb = −Aa(g00 + 1). (A.3)
We are interested in the low-energy behavior of this system. Therefore, following the reason-ing of section 4, we disregard the time derivative ∂2
t + 3H∂t when compared to the operatorM2
a −∇2. On the other hand, we do not neglect the time derivative in the interaction term,as its role is to couple different massive fields, and its contribution depends on the strengthof Cab. These considerations lead to the equation
ΩaFa +∑
b
CabFb = −Aa(g00 + 1), (A.4)
whereΩa ≡ M2
a −∇2 −Ba(g00 + 1). (A.5)
Since in this limit the heavy fields Fa are non-dynamical, we may treat them as Lagrangemultipliers and insert them back into the action without kinetic terms. This leads to theEFT action contribution due to the heavy fields:
S = −1
2
∫
d3xdt∑
a
(g00 + 1)AaFa, (A.6)
where the Fa are the solutions of (A.4) which we now proceed to obtain. First, noticethat (A.4) may be re-expressed as:
Ω1 C12∂t C13∂t · · ·−C12∂t Ω2 C23∂t · · ·−C13∂t −C23∂t Ω3 · · ·
......
.... . .
F1
F2
F3...
= −
A1
A2
A3...
(g00 + 1). (A.7)
To deal with this equation, we assume that the off-diagonal terms are subleading when com-pared to the diagonal terms Ωa. This allows us to invert the matrix operator perturbatively,leading to the first-order result:
F1
F2
F3...
= −
Ω−11 −Ω−1
1 C12∂tΩ−12 −Ω−1
1 C13∂tΩ−13 · · ·
Ω−12 C12∂tΩ
−11 Ω−1
2 −Ω−12 C23∂tΩ
−13 · · ·
Ω−13 C13∂tΩ
−11 Ω−1
3 C23∂tΩ−12 Ω−1
3 · · ·...
......
. . .
A1
A2
A3...
(g00+1), (A.8)
which may be re-expressed as:
Fa = −Aa
Ωa(g00 + 1) +
∑
b
Cab1
Ωa∂t
1
ΩbAb(g
00 + 1). (A.9)
We may now plug this solution back into the action, obtaining:
S=1
2
∫
d3xdt
∑
a
AaAa(g00+1)
1
Ωa(g00+1)−
∑
ab
AaCab(g00+1)
1
Ωa∂t
1
ΩbAb(g00+1)
. (A.10)
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JCAP04(2013)004
To simplify this expression notice that, due to the anti-symmetry of Cab, the second termvanishes whenever the time derivative ∂t acts on a quantity that does not carry the labelb. This means that the only non-vanishing contributions coming from the second term arethose proportional to Ab, Bb and M2
b . For definiteness, and to keep our discussion simple,let us assume that both Bb and M2
b are constants and consider only a time dependence ofthe Aa coefficients. In this case, we obtain the formal result:
S=1
2
∫
d3xdt
∑
a
AaAa(g00+1)
1
Ωa(g00+1)−
∑
ab
(CabAaAb)(g00+1)
1
ΩaΩb(g00+1)
. (A.11)
As discussed in section 4, the inverse of Ωa is an operator which has the following expansion:
Ω−1a =
1
M2a −∇2
[
1− (g00 + 1)Ba
M2a −∇2
]−1
=1
M2a −∇2
∑
n
[
(g00 + 1)Ba
M2a −∇2
]n
. (A.12)
Then, inserting this expansion back into the action (A.11) and keeping terms up to cubicorder, we finally arrive at the expression
S =1
2
∫
d3xdt
(g00 + 1)
[
∑
a
A2a
M2a −∇2
−∑
ab
CabAaAb
(M2a −∇2)(M2
b −∇2)
]
(g00 + 1) (A.13)
+∑
a
A2aBa(g
00 + 1)1
M2a −∇2
[
(g00 + 1)1
M2a −∇2
(g00 + 1)
]
−∑
ab
CabAaAbBb(g00 + 1)
1
M2a −∇2
[
(g00 + 1)1
(M2b −∇2)(M2
c −∇2)(g00 + 1)
]
−∑
ab
CabAaAbBa(g00 + 1)
1
(M2b −∇2)(M2
c −∇2)
[
(g00+1)1
M2a − ∇2
(g00+1)
]
+ · · ·
.
This implies that the general quadratic action for the Goldstone boson π takes the form
S(2) = −M2Pl
∫
d3xdta3H
[
π
(
1 +∑
a
βa
M2a − ∇2
+∑
ab
βab
(M2a − ∇2)(M2
b − ∇2)+ · · ·
)
π
−(∇π)2]
, (A.14)
where βa parametrizes the coupling to a heavy field with index a, and βab parametrize theinteractions between heavy fields carrying labels a and b etc. In momentum space the actiontakes the form
S(2) = −M2Pl
∫
d3kdta3H
[
π
(
1 +∑
a
βaM2
a + p2+∑
ab
βab(M2
a + p2)(M2b + p2)
+ · · ·)
π
+p2π2
]
. (A.15)
The equation of motion for the π field is therefore given by
π + 3Hπ − c2s (p2)p2π = 0, (A.16)
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JCAP04(2013)004
where
c2s (p) =∏
a
(M2a + p2)× (A.17)
[
∏
a
(M2a + p2)+
∑
a
βa∏
b 6=a
(M2b +p2)+
∑
a<b
βab∏
c 6=a,b
(M2c +p2)+. . .+β12...N
]−1
.
The inverse speed of sound squared is defined as the limit
c−2s ≡ lim
p→0c−2s (p) = 1 +
∑
a
βaM2
a
+∑
a<b
βabM2
aM2b
+ . . .+β12...N
M21M
22 . . .M
2N
, (A.18)
where N is the number of heavy fields and the indices run from 1 . . . N . To analyze this,let us consider the short wavelength regime where the friction term can be disregarded andp ≡ k/a may be taken as a constant. The dispersion relation is then
ω2(p) = c2s (p)p2. (A.19)
For the case of one additional heavy field we get
c2s (p) = (M2 + p2)[
M2 + p2 + β]−1
, (A.20)
which reduces to the expressions in eq. (3.9), (3.10) when β =2M4
2M2
M2Pl|H|
. For multiple non-
interacting fields where βab... = 0, this becomes
c2s (p) =∏
a
(M2a + p2)
[
∏
a
(M2a + p2) +
∑
a
βa∏
b 6=a
(M2b + p2)
]−1
, (A.21)
with the inverse speed of sound squared given by
c−2s = 1 +
∑
a
βaM2
a
. (A.22)
Recall that we are restricted to the low-energy regime
ω2 ≪ M2a + p2
in order for the expansion (A.5) to be valid. Without loss of generality we can consider twocases: one where the Ma are all comparable, and the other where there exists some hierarchyamong these heavy masses. This can be studied using a representative lowest mass M2
l ;either other masses are comparable, or significantly larger. In the former case we require theinequality to hold for all a, while in the latter just that ω2 ≪ M2
l + p2. The generic UVscale for arbitrary number of fields with different masses will be a complicated function ofthe speed of sound and the mass scales of the problem, so let us study in some detail thecase where all the heavy masses Ma are comparable: M2
a ≈ M2 ∀ a. The dispersion relationthen reads
ω2(p) = (M2 + p2)p2 × (A.23)[
M2 + p2 +∑
a
βa + (M2 + p2)−1∑
a<b
βab + . . .+ β12...N (M2 + p2)1−N
]−1
.
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JCAP04(2013)004
From this expression we can read off the low-energy regime as an upper bound in themomentum
p2 ≪ M2 +∑
a
βa + (M2 + p2)−1∑
a<b
βab + . . .+ β12...N (M2 + p2)1−N . (A.24)
We see that in general this is a polynomial inequality of degree N in squared momentum,
GN (p2) ≪ 0 .
Therefore the solution is p2 ≪ p2UV(M, cs, β) with p2UV representing the degenerate positiveroot of the polynomial GN . The energy scale ΛUV is then given by substituting p2UV intothe dispersion relation. Since this is the root of the polynomial GN (p2) the denominator ofeq. (A.23) is just proportional to p2UV and the expression simplifies to
Λ2UV ∼ (M2 + p2UV) . (A.25)
We also see a modification of the dispersion relation in the multiple heavy field case.For small values of p2 compared to the mass squared, the low-energy regime condition (A.24)becomes
p2 ≪ M2 +∑
a
βa +∑
a 6=b
βabM−2 + . . .+ β12...nM
2(1−n). (A.26)
This inequality is automatically satisfied when p2 ≪ M2. The dispersion relation in thisregime becomes
ω2(p) = c2sp2
(
1 +p2
M2
)n
, (A.27)
where c2s is given eq. (A.18). Note that the sound speed is lowered, and lowered additively, bythe presence of heavy fields in this regime. The expansion (A.27) includes terms dependenton p4, p6 . . ., but these are suppressed by increasing powers of p2/M2, so that we recover theusual ω2 ∼ p2 dispersion relation in this regime. For large values of p2 compared to M2, thelow-energy condition becomes
p2n ≪(
∑
a
βap2(n−1) +
∑
a 6=b
βabp2(n−2) + . . .+ β123...n
)
. (A.28)
The dispersion relation in this regime is given by
ω2(p) = (∑
βap−4 +
∑
βabp−6 + . . .+ β1...np
−2n−2)−1. (A.29)
We see that many powers of p can enter. However, for large p2 ≫ M2, the subleading termsin Λ(p) in the denominator are suppressed, and the dominant p-dependence of the dispersionrelation is given by
ω2(p) ≈ p4∑
βa. (A.30)
– 28 –
JCAP04(2013)004
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